Monomial orderings, rewriting systems, and Gröbner bases for the ...

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$Math 905 Solutions to Homework # 1 Throughout R denotes a ...$

e the canonical projection with kernel the commutator ideal I. In this section we will prove the following theorem. Theorem 3.1 Let < be a term ordering on K[x, y, z]. There exist mono- ) ⊂ R, mial orderings ≺r on A = K〈a, b, c〉, for each r in the interval (0, 1 2 which are liftings of
First consider the case that there are only finitely many such indices mj, with mJ being the largest one. Set mj = ∞, if j > J. In this case we have rules (4 j ) for 1 ≤ j ≤ J + 1 and rules (5 j ) for 1 ≤ j ≤ J. Note that in this case, the rewriting system has the additional property that it is regular, in the sense that the set of word pairs given by the left and right hand sides of rules is a regular language accepted by a finite state automaton. The normal forms associated to the rewriting system are therefore also a regular language. To each word w in Σ ∗ we associate an integer vector Ψ(w) as follows. Each of the entries of this vector Ψ(w) will be a “barrier ordering”; we will place barriers in w, and then use these barriers to associate a number to w. To define the first entry α(w) of Ψ(w), insert barriers in w to immediate the right of every letter c that occurs. For instance, if w = abc 2 baca, then we place barriers as follows: w = abc|c|bac|a. Let lb(i) denote the number of occurrences of b to the left of the i-th barrier in w. Suppose there are n barriers in w. Now define α(w) = lb(1) + · · · + lb(n). For the example w = abc 2 baca above, we have α(w) = 1 + 1 + 2 = 4. The other entries of Ψ(w) are defined similarly, but the barriers will be placed after certain types of subwords rather than individual letters. First we introduce some notation. Given any word t in Σ ∗ , let ˜t be the word obtained from t by deleting all occurrences of a in t, and let la(t) denote the number of occurrences of a in t. Suppose s is a word involving just the letters b and c, and w is a word in Σ ∗ as above. We can write w = v1s1v2s2...skvk+1 where: (i) ˜si = s, (ii) if ti is a proper subword of si then ˜ti = s, and (iii) k is maximal. 7
Page 1 and 2: Monomial orderings, rewriting syste
Page 3 and 4: monomials in the free algebra are c
Page 5: A rewriting system (Σ, R) is termi
Page 9 and 10: Then We may write w as w = c|ac|b 2
Page 11 and 12: Next we show that the reduction pro
Page 13: [2] E. Green, T. Mora and V. Ufnaro

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